Quantum Fluctuation Theorems and Work–Energy Relationships with Due Regard for Convergence, Dissipation and Irreversibility

Abstract. In this article, firstly the fluctuation theorems (FT)
for expended work in a driven nonequilibrium system, isolated or thermostatted, as
formulated originally by Crooks and Tasaki, together with the ensuing
Jarzynski work–energy (W–E) relationships, will be discussed and
reobtained. Secondly, the fluctuation theorems for entropy flow due to Evans, Cohen and
Morriss with extensions by many researchers, a.o. Evans and Searles,
Gallavotti and Cohen, Kurchan, Lebowitz and Spohn, and Harris and Schütz
will be reconsidered. Our treatment will be fully quantum-statistical, being
an extension of our previous research reported in Phys. Rev. E, 2012. While
a true explosion of papers took place after the initial articles at the turn
of the century, virtually all of these suffered from one or more of the
following deficiencies: (i). The arguments are based on classical
trajectories in phase space; this is true for Christopher Jarzynski's
original work, as well as for Crooks' paper; better fares Tasaki's quantum
paper in the arXiv. (ii). Many quantum treatments involve the 'pure' von
Neumann equation or 'non-reduced' Heisenberg operators. This is regrettable
particularly for an otherwise beautiful derivation in the complex plane by
Talkner and Hänggi; correlation functions for non-reduced Heisenberg
operators do not converge. As we pointed out in many papers and in our
recent book: Kubo Linear Response Theory (LRT) is a hollow shell until proper
randomization (Kubo: 'stochasticization') is introduced and carried out.
Hence, the interactions λσ with the reservoir or
internal causes must explicitly be considered. Taking the trace over these, the resulting semigroup has complete positivity, exhibits non-unitarity for
the time evolution, dissipation and irreversibility, the general
result being the Lindblad quantum master equation (QME). In the physical
literature a more explicit result is obtained after application of the 'weak
coupling–long time' limit, developed long ago by Leon Van Hove. In our
cited paper these results have been extended to non-stationary processes,
the result being concordant with work by Gaspard. (iii). Whereas a few dozen
papers use a stochastic approach with some Master Equation as leitmotiv, this author
found most treatments wanting and not in accord with the general tenets
spelled out by Lindblad and others, e.g. Breuer and Petruccione. In
particular, a stochastic treatment with 'jump-induced' random trajectories
as by Harris and Schütz is begging the question. While a number of their
relationships will still be employed, our Markov probability P(σf
,tf∣σ0 ,t0) shall only denote the two state-points,
with no reference whatsoever to stochastic trajectories, these being
meaningless in a quantum description. A straightforward reasoning gives the
desired results.